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Properties

Label 2-3087-21.20-c1-0-21
Degree $2$
Conductor $3087$
Sign $-0.577 - 0.816i$
Analytic cond. $24.6498$
Root an. cond. $4.96485$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.98i·2-s − 1.94·4-s − 3.55·5-s + 0.0996i·8-s − 7.06i·10-s − 4.53i·11-s − 4.71i·13-s − 4.09·16-s − 7.51·17-s − 1.30i·19-s + 6.92·20-s + 9.01·22-s − 0.567i·23-s + 7.62·25-s + 9.37·26-s + ⋯
L(s)  = 1  + 1.40i·2-s − 0.974·4-s − 1.58·5-s + 0.0352i·8-s − 2.23i·10-s − 1.36i·11-s − 1.30i·13-s − 1.02·16-s − 1.82·17-s − 0.298i·19-s + 1.54·20-s + 1.92·22-s − 0.118i·23-s + 1.52·25-s + 1.83·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3087\)    =    \(3^{2} \cdot 7^{3}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(24.6498\)
Root analytic conductor: \(4.96485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3087} (3086, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3087,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8156425580\)
\(L(\frac12)\) \(\approx\) \(0.8156425580\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 1.98iT - 2T^{2} \)
5 \( 1 + 3.55T + 5T^{2} \)
11 \( 1 + 4.53iT - 11T^{2} \)
13 \( 1 + 4.71iT - 13T^{2} \)
17 \( 1 + 7.51T + 17T^{2} \)
19 \( 1 + 1.30iT - 19T^{2} \)
23 \( 1 + 0.567iT - 23T^{2} \)
29 \( 1 - 8.53iT - 29T^{2} \)
31 \( 1 - 7.69iT - 31T^{2} \)
37 \( 1 - 3.89T + 37T^{2} \)
41 \( 1 - 7.52T + 41T^{2} \)
43 \( 1 - 4.79T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 4.85iT - 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 - 0.664iT - 61T^{2} \)
67 \( 1 - 5.93T + 67T^{2} \)
71 \( 1 + 4.70iT - 71T^{2} \)
73 \( 1 - 5.57iT - 73T^{2} \)
79 \( 1 + 1.46T + 79T^{2} \)
83 \( 1 - 2.54T + 83T^{2} \)
89 \( 1 - 3.60T + 89T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.490227233436257088021494076923, −8.260560809939314576481565983730, −7.40383079052376505431511402258, −6.85817447001897484119258942410, −6.09433005945727784557439736986, −5.20761903634016669484345197984, −4.54173459477850635380725995944, −3.57670968466406422229406043208, −2.75503013715085451185630639786, −0.65026048352121804475745330531, 0.42882355512703784941945808946, 1.95741111161194899581341300414, 2.50342572754317676532722429597, 3.90871550402200578056576858713, 4.19211197769524076956852381192, 4.67959920175999227354074772824, 6.39111407395385253787941407056, 7.03457479456026000412524884942, 7.75491401827675148063627197195, 8.589311605944476870564584339090

Graph of the $Z$-function along the critical line